Homology classes of real algebraic sets
The program of comparison between algebraic and topological categories to which the present paper contributes has its origin in \textit{J. Nash}'s celebrated 1952 paper on real algebraic manifolds [Ann. Math. (2) 56, 405--421 (1952; Zbl 0048.38501)]. It obtained a strong impetus by \textit{A. Tognoli}'s 1973 paper [Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 27, 67--185 (1973; Zbl 0263.57011)] where it is proved that every compact \(C^\infty\) submanifold of an \(\mathbb{R}^m\) is diffeomorphic to a nonsingular algebraic subset of an \(\mathbb R^p;\) that is, every such manifold has a so called algebraic model. For this see also chapter 14 of the book [BCR] by [\textit{J. Bochnak, M. Coste, M.-F. Roy}, Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 36. (Berlin): Springer. (1998; Zbl 0912.14023)]. In the following we closely follow the author's introduction making use of [BCR]. Given an affine real algebraic variety \(X\) in the sense of [BCR, p. 63], denote by \(H_p(X,\mathbb{Z}/2)\) its \(p\)-th homology group with coefficients in \(\mathbb{Z}/2,\) by \(H_p^{\text{alg}}(X,\mathbb{Z}/2)\) the subgroup presented by the \(p\)-dimensional subvarieties of \(X,\) and let \(H^q_{\text{alg}}(X,\mathbb{Z}/2)\) be its inverse image under Poincaré Duality. So \(p+q=n.\) The groups \(H^q_{\text{alg}}(-,\mathbb{Z}/2)\) of algebraic cohomology classes are central in real algebraic geometry and satisfy the expected functorial property that given a regular map \(f:X\rightarrow Y\) between compact nonsingular real algebraic varieties then the induced homomorphism \(f^*: H^q(Y,\mathbb Z/2)\rightarrow H^q(X,\mathbb Z/2)\) maps the subgroup \(H^q_{\text{alg}}(Y,\mathbb{Z}/2)\) into \(H^q_{\text{alg}}(X,\mathbb{Z}/2).\) The direct sum \(H^*_{\text{alg}}(X,\mathbb{Z}/2)= \bigoplus_{q\geq 0} H^q_{\text{alg}}(X,\mathbb{Z}/2)\) is a subring of the cohomology ring sum \(H^*(X,\mathbb{Z}/2)\) and the Stiefel-Whitney class \(w_q(X)\)is in \(H^q_{\text{alg}}(X,\mathbb{Z}/2),\) see [BCR, p. 304]. The set \(\text{Alg}^q(X)\) of all cohomology classes \(u\) in \(H^q_{\text{alg}}(X,\mathbb{Z}/2)\) that are algebraically equivalent to \(0\) forms a subgroup of this group; in fact \({\text{Alg}}^*(X)\) is an ideal in \(H^*_{\text{alg}}(X,\mathbb{Z}/2).\) Given a nonsingular subvariety \(Y\) of \(X,\) the groups \(H^1_{\text{alg}}(Y,\mathbb{Z}/2)\) and \({\text{Alg}}^1(Y)\) are \textit{determined by \(X\)} if applying the homomorphism \(i^*\) induced from inclusion to \(H^1_{\text{alg}}(X,\mathbb{Z}/2)\) and \(\text{Alg}^1(X)\) one obtains the first ones. Whether this is the case is often hard to decide. To make progress the author introduces the concept of \textit{movability} of \(Y\) in \(X\) which means roughly speaking that there are sufficiently many displacements \(Y_t\) of \(Y\) such that \(H^1_{\text{alg}}(Y_t,\mathbb{Z}/2)\) and \(\text{Alg}^1(Y_t)\) are determined by \(X.\) Here the \(t\) lie in some \(\mathbb{R}^k.\) It is shown that movable subvarieties of \(X\) occur in a natural way. T 1.4. Let \(\xi\) be and algebraic vector bundle on \(X\) with 2+rank\(\xi\leq \dim X.\) If \(s:X \rightarrow \xi\) is an algebraic section transverse to the zero section then the nonsingular subvariety \(Y=s^{-1}(0)\) of \(X\) is movable. Algebraic vector bundles as used here are the topic of [BCR], chapter 12. A smooth submanifold \(M\) of \(X\) is \textit{admissible} if arbitrarily near to the identity there is a diffeomorphism \(\varphi\) of \(X\) such that \(Y=\varphi(M)\) is an irreducible nonsingular subvariety of \(X\) whose groups \(H^1_{\text{alg}}(Y,\mathbb{Z}/2)\) and \(\text{Alg}^1(Y)\) are determined by \(X.\) Using that the algebraic sections of an algebraic vector bundle \(\xi\) over \(X\) are dense in the space of all \(C^\infty\) sections of \(\xi,\) the following corollary is obtained C1.7: With \(\xi,X\) as above, if \(\sigma:X\rightarrow \xi\) is a smooth section transverse to the zero section, then the smooth manifold \(M=\sigma^{-1}(0)\) of \(X\) is admissible. The author then examines, given an arbitrary subvariety \(Y\) of \(X\) what are the relations between the triples \((H^1(X,\mathbb{Z}/2), H^1_{\text{alg}}(X,\mathbb{Z}/2), \text{Alg}^1(X))\) and the corresponding one for \(Y.\) Given manifold \(M\) and a submanifold \(N\) of lower dimension dim\(N\geq 3,\) both connected smooth, and \(M\) compact, and subgroups \(\Gamma_M\subseteq G_M\) of \(H^1(M,\mathbb{Z}/2)\) and \(\Gamma_N\subseteq G_N\) of \(H^1(N,\mathbb{Z}/2),\) a necessary and sufficient condition for the existence of real algebraic variety \(X,\) a nonsingular subvariety \(Y\) of \(X\) and a smooth diffeomorphism \(\varphi:X\rightarrow M\) such that \(\varphi(Y)=N\) and the equations \(\varphi^*(G_M)=H^1_{\text{alg}}(X,\mathbb{Z}/2),\) \( \varphi^*(\Gamma_M)=\text{Alg}^1(X), \) \((\varphi_{|Y})^*(G_N)=H^1_{\text{alg}}(Y,\mathbb{Z}/2),\) \((\varphi_{|Y})^*(\Gamma_N)=\text{Alg}^1(Y), \) hold true is given. This condition involves the graded subring of the cohomology ring generated by Stiefel-Whitney classes of manifolds \(M,N.\) A further result, apparently independent of the bulk of this paper is a generalization of the counterexample of \textit{R. Benedetti} and \textit{M. Dedò} [Compos. Math. 53, 143--151 (1984; Zbl 0547.14019)] that was to be a major step towards a topological characterization of all real algebraic sets.
scientific article; zbMATH DE number 5298328
scientific article